On (N,M)-a-normal And (N, M)-a-quasinormal Semi- Hilbert Space Operators (original) (raw)
Positive-Normal Operators in Semi-Hilbertian Spaces
International Mathematical Forum, Vol. 9, 2014, no. 11, 533 - 559, 2014
Given a bounded positive linear operator A on a Hilbert space H we consider the semi-Hilbertian space (H , | A), where ξ | η A := Aξ | η. In this paper we introduce a class of operators on a semi Hilbertian space H with inner product | A. We call the elements of this class A-positive-normal or A-posinormal. An operator T ∈ B(H) is said to be A-posinormal if there exists a A-positive operator P ∈ B(H) (i.e., AP ≥ 0) such that T AT * = T * AP T. We study some basic properties of these operators. Also we study the relationship between a special case of this class with the other kinds of classes of operators in semi-Hilbertian spaces.
A-m-Isometric operators in semi-Hilbertian spaces
Linear Algebra and its Applications, 2012
In this work, the concept of m-isometry on a Hilbert space are generalized when an additional semi-inner product is considered. This new concept is described by means of oblique projections.
Orthogonality and norm attainment of operators in semi-Hilbertian spaces
Annals of Functional Analysis, 2020
We study the semi-Hilbertian structure induced by a positive operator A on a Hilbert space ℍ. Restricting our attention to A−bounded positive operators, we characterize the norm attainment set and also investigate the corresponding compactness property. We obtain a complete characterization of the A−Birkhoff-James orthogonality of A−bounded operators under an additional boundedness condition. This extends the finite-dimensional Bhatia-Semrl Theorem verbatim to the infinite-dimensional setting.
Operators on positive semidefinite inner product spaces
Linear Algebra and its Applications, 2020
Let U be a semiunitary space; i.e., a complex vector space with scalar product given by a positive semidefinite Hermitian form x¨,¨y. If a linear operator A : U Ñ U is bounded (i.e., }Au} ď c}u} for some c P R and all u P U), then the subspace U 0 :" tu P U | xu, uy " 0u is invariant, and so A defines the linear operators A 0 : U 0 Ñ U 0 and A 1 : U{U 0 Ñ U{U 0. Let A be an indecomposable bounded operator on U such that 0 ‰ U 0 ‰ U. Let λ be an eigenvalue of A 0. We prove that the algebraic multiplicity of λ in A 1 is not less than the geometric multiplicity of λ in A 0 , and the geometric multiplicity of λ in A 1 is not less than the number of Jordan blocks J t pλq of each fixed size tˆt in the Jordan canonical form of A 0. We give canonical forms of selfadjoint and isometric operators on U, and of Hermitian forms on U. For an arbitrary system of semiunitary spaces and linear mappings on/between them, we give an algorithm that reduces their matrices to canonical form. Its special cases are Belitskii's and Littlewood's algorithms for systems of linear operators on vector spaces and unitary spaces, respectively.
Refinement of seminorm and numerical radius inequalities of semi-Hilbertian space operators
arXiv: Functional Analysis, 2020
We give new inequalities for AAA-operator seminorm and AAA-numerical radius of semi-Hilbertian space operators and show that the inequalities obtained here generalize and improve on the existing ones. Considering a complex Hilbert space mathcalH\\mathcal{H}mathcalH and a non-zero positive bounded linear operator AAA on mathcalH,\\mathcal{H},mathcalH, we show with among other seminorm inequalities, if S,T,XinmathcalBA(mathcalH)S,T,X\\in \\mathcal{B}_A(\\mathcal{H})S,T,XinmathcalBA(mathcalH), i.e., if AAA-adjoint of S,T,XS,T,XS,T,X exist then 2\\|S^{\\sharp_A}XT\\|_A \\leq \\|SS^{\\sharp_A}X+XTT^{\\sharp_A}\\|_A.$$ Further, we prove that if TinmathcalBA(mathcalH)T\\in \\mathcal{B}_A(\\mathcal{H})TinmathcalBA(mathcalH) then \\begin{eqnarray*} \\frac{1}{4}\\|T^{\\sharp_{A}}T+TT^{\\sharp_{A}}\\|_A \\leq \\frac{1}{8}\\bigg( \\|T+T^{\\sharp_{A}}\\|_A^2+\\|T-T^{\\sharp_{A}}\\|_A^2\\bigg), ~~\\textit{and} \\end{eqnarray*} \\begin{eqnarray*} \\frac{1}{8}\\bigg( \\|T+T^{\\sharp_{A}}\\|_A^2+\\|T-T^{\\sharp_{A}}\\|_A^2\\bigg) +\\frac{1}{8}c_A^2\\big(T+T^{\\sharp_{A}}\\big)+\\frac{1}{8}c_A^2\\big(T-T^{\\sharp_{A}}\\big) \\leq w^2_A(T). \\end{eqnarray*} Here wA(.),cA(.)w_A(.), c_A(.)wA(.),cA(.) ...
DergiPark (Istanbul University), 2021
in [1]. In this paper we introduce a new classes of operators on semi-Hilbertian space (ℋ, ∥. ∥) called (,) power-(,)-normal denoted [(,) ] and (,) power-(,)-quasi-normal denoted [(,) ] associated with a Drazin invertible operator using its Drazin inverse. Some properties of [(,) ] and [(,) ] are investigated and some examples are also given. An operator ∈ ℬ (ℋ) is said to be (n, m) power-(,)normal for some positive operator and for some positive integers and if () (⋕) = (⋕) () .
Semi-inner products and operators which attain their norm
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2021
It has been proved by Koehler and Rosenthal [Studia Math. 36 (1970), 213-216] that an linear isometry U 2 LðXÞ preserves some semi-inner-product. Recently, similar investigations have been carried out by Niemiec and Wójcik for continuous representations of amenable semigroups into LðXÞ (cf. [Studia Math. 252 (2020), 27-48]). In this paper we generalize the result of Koehler and Rosenthal. Namely, we prove that if an operator T 2 LðXÞ of norm one attains its norm then there is a semi-inner-product ½ÁjÅ : X  X ! F that the operator T preserves this semi-inner-product on the norm attaining set. More precisely, we show that the equality ½T ðÁÞjT x ¼ ½Ájx holds for all x 2 M T :¼ fy 2 S X : kT yk ¼ 1g.
On approximate orthogonality and symmetry of operators in semi-Hilbertian structure
Bulletin des Sciences Mathématiques, 2021
The purpose of the article is to generalize the concept of approximate Birkhoff-James orthogonality, in the semi-Hilbertian structure. Given a positive operator A on a Hilbert space H, we define (ǫ, A)−approximate orthogonality and (ǫ, A)−approximate orthogonality in the sense of Chmieliński and establish a relation between them. We also characterize (ǫ, A)−approximate orthogonality in the sense of Chmieliński for Abounded and Abounded compact operators. We further generalize the concept of right symmetric and left symmetric operators on a Hilbert space. The utility of these notions are illustrated by extending some of the previous results obtained by various authors in the setting of Hilbert spaces.
On operators satisfying an inequality
Journal of Inequalities and Applications, 2012
,B ∈ B(B(H)) denote either the generalized derivation δ A,B = L A -R B or the elementary operator A,B = L A R B -I, where L A and R B are the left and right multiplication operators defined on B(H) by L A = AX and R B = XB respectively. This article concerns some spectral properties of k-quasi- * -class A operators in a Hilbert space, as the property of being hereditarily polaroid. We also establish Weyl-type theorems for T and d A,B , where T is a k-quasi- * -class A operator and A, B * are also k-quasi- * -class A operators. MSC: Primary 47B47; 47A30; 47B20; secondary 47B10